[FOM] Only one proof
Aatu Koskensilta
Aatu.Koskensilta at uta.fi
Fri Sep 4 01:25:11 EDT 2009
Quoting John Baldwin <jbaldwin at uic.edu>:
> On Mon, 31 Aug 2009, Aatu Koskensilta wrote:
>
>> Another example: Gödel's proof for the first incompleteness theorem
>> understood as the result that the set of Pi-1 truths is productive.
>> I'm not aware of any way of proving this result without going through
>> essentially the recursion theoretic contortions found in the original
>> proof. (I may well be just ignorant.)
>
> I am not sure of how `understood as productive' affects the
> observation that via Paris-Harrington, the first incompleteness
> theorem can be proved without coding syntax. This sure qualifies as
> a different proof.
A different proof, to be sure. But how do you propose to squeeze out
of Paris-Harrington the productivity of the set of Pi-1 truths?
Charles Silver mentions also Kripke's model theoretic proof (of Pi-2
incompleteness). This proof too doesn't yield in any obvious way the
productivity of the set of Pi-1 truths -- which, arguably, is the
recursion theoretic core of the first incompleteness theorem.
--
Aatu Koskensilta (aatu.koskensilta at uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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